Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23866 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,19)(15,16,20,22)(17,25,27,21)(23,26,28,24);(5,7,9,8)(6,10,12,11)(13,15)(14,16)(17,26)(18,20)(19,22)(21,23)(24,27)(25,28);(2,4)(5,7,9,8)(6,11,12,10)(14,19)(15,22)(16,20)(17,27)(23,24)(26,28);(5,8,9,7)(6,10,12,11)(13,16,18,22)(14,20,19,15)(17,23,27,28)(21,24,25,26);(13,17,18,27)(14,21,19,25)(15,23,20,28)(16,24,22,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484373);